Video Transcript
True or False: 26 minus the
difference between 15 and the square root of seven is equal to the difference
between 26 and 15 minus the square root of seven. Is subtraction associative or
nonassociative in the set of real numbers? True or False: The square root of
seven minus 15 is equal to 15 minus the square root of seven. Is subtraction commutative or
noncommutative in the set of real numbers?
In the first part of this question,
we are asked to determine if an equation holds true. One way of doing this is to try and
rewrite both sides of the equation to be in the same form. To do this, let’s start by
distributing the negative over the terms inside the parentheses on the left-hand
side of the equation. This gives us 26 minus 15 plus the
square root of seven. On the right-hand side of the
equation, we can evaluate the difference between the integers inside the parentheses
to obtain 11 minus the square root of seven.
We can simplify the expression on
the left-hand side of the proposed equation to get 11 plus the square root of
seven. We can see that this is not the
same as 11 minus the square root of seven since the left-hand side is greater than
11 and the right-hand side is less than 11. Hence, the two sides of the
proposed equation are not equal. So the answer is false.
In this second part of this
question, we want to determine whether subtraction is an associative or
nonassociative operation over the set of real numbers. We can recall that for an operation
to be associative, we must be able to evaluate the operation in any order. So, we must have that for any real
numbers 𝑎, 𝑏, and 𝑐, 𝑎 minus the difference between 𝑏 and 𝑐 is equal to the
difference between 𝑎 and 𝑏 minus 𝑐. However, we have shown in the first
part of this question that this does not hold true for all real values. In particular, it does not hold
true when 𝑎 is equal to 26, 𝑏 is equal to 15, and 𝑐 is equal to the square root
of seven. Hence, we have shown that the
subtraction operation is nonassociative over the set of real numbers.
In the third part of this question,
we need to determine if two sides of a proposed equation are equal. We can see that the two expressions
are the same two real numbers. However, the order of the numbers
has been reversed in the subtraction. There are many ways of showing that
the two sides of the equation are not equal. One way is to note that the square
root of seven lies between two and three, since two squared is less than seven, but
three squared is greater than seven. Therefore, the square root of seven
minus 15 will be less than zero, and 15 minus the square root of seven will be
greater than zero. Since the left-hand side of the
proposed equation is negative and the right-hand side is positive, they cannot be
equal. So the answer is false.
In the final part of this question,
we want to determine if the subtraction operation is commutative or noncommutative
over the set of real numbers. We can recall that for an operation
to be commutative, we need to be able to reorder the elements in the operation. Therefore, for subtraction to be
commutative over the set of real numbers, we need 𝑎 minus 𝑏 to be equal to 𝑏
minus 𝑎 for all real numbers 𝑎 and 𝑏. However, we have already shown that
this is not true when 𝑎 is equal to the square root of seven and 𝑏 is equal to
15. Hence, we have shown that the
subtraction operation is noncommutative over the set of real numbers.
